meromorphic

mero- +‎ -morphic

meromorphic (not comparable)

1. (complex analysis, of a function) That is the ratio of two holomorphic functions (and so possibly infinite at a discrete set of points).
   • 1993, Joel L. Schiff, Normal Families, Springer, page 71,

     Normal families of meromorphic functions are most naturally studied using the spherical metric (§1.2), an approach initiated by Ostrowski [1926]. Some results for meromorphic functions, such as the FNT, are immediate extensions from the analytic case, whereas others, such as Landau's or Julia's theorem are set in a much broader context than their analytic counterparts. Normality criteria pertinent to families of meromorphic functions, such as Marty's theorem, have not yet been encountered.

   • 2000, Werner Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Springer, page 39,

     Note that such a transformation is holomorphic at the origin, but is essentially singular at infinity. However, since ${\displaystyle T(z)}$ commutes with ${\displaystyle A(z)}$ , the transformed system has coefficient matrix ${\displaystyle A(z)-zq'(z)I}$ and hence is again meromorphic at infinity.

   • 2012, Marius van der Put, Michael F. Singer, Galois Theory of Linear Differential Equations, Springer, page 147,

     A point ${\displaystyle p\in P^{1}}$ is singular for ${\displaystyle \textstyle {\frac {d}{dz}}-A}$ if the equation cannot be made regular at ${\displaystyle p}$ with a local meromorphic transformation.

• holomorphic
• meromorphism